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arXiv:1606.00960 · Physical Review A

Projecting three-dimensional color codes onto three-dimensional toric codes

Arun B. Aloshious, Pradeep Kiran Sarvepalli

Published Jul 2, 2018• 16 citations
arXivHTMLPDF

Abstract

Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica et al., New J. Phys. 17, 083026 (2015)] showed that any $D$-dimensional color code can be mapped to a finite number of toric codes in $D$ dimensions. We propose an alternate map of three-dimensional (3D) color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result [N. Delfosse, Phys. Rev. A 89, 012317 (2014)] for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit-flip errors are decoded by projecting on one set of 3D toric codes, while phase-flip errors are decoded by projecting onto another set of 3D toric codes. We use these projections to study the performance of a class of 3D color codes called stacked codes.

DOI
10.1103/physreva.98.012302

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Toric codes and color codes are two important classes of topological codes. Kubica et al. [A. Kubica et al., New J. Phys. 17, 083026 (2015)] showed that any DDD-dimensional color code can be mapped to a finite number of toric codes in DDD dimensions. We propose an alternate map of three-dimensional (3D) color codes to 3D toric codes with a view to decoding 3D color codes. Our approach builds on Delfosse's result [N. Delfosse, Phys. Rev. A 89, 012317 (2014)] for 2D color codes and exploits the topological properties of these codes. Our result reduces the decoding of 3D color codes to that of 3D toric codes. Bit-flip errors are decoded by projecting on one set of 3D toric codes, while phase-flip errors are decoded by projecting onto another set of 3D toric codes. We use these projections to study the performance of a class of 3D color codes called stacked codes.
Three dimensional (3D) toric codes are a class of stabilizer codes with local checks and come under the umbrella of topological codes. While decoding algorithms have been proposed for the 3D toric code on a cubic lattice, there have been very few studies on the decoding of 3D toric codes over arbitrary lattices. Color codes in 3D can be mapped to toric codes. However, the resulting toric codes are not defined on cubic lattice. They are arbitrary lattices with triangular faces. Decoding toric codes over an arbitrary lattice will help in studying the performance of color codes. Furthermore, gauge color codes can also be decoded via 3D toric codes. Motivated by this, we propose an efficient algorithm to decode 3D toric codes on arbitrary lattices (with and without boundaries).
Recent years have seen a growing interest in quantum codes in three dimensions (3D). One of the earliest proposed 3D quantum codes is the 3D toric code. It has been shown that 3D color codes can be mapped to 3D toric codes. The 3D toric code on a cubic lattice is also a building block for the welded code which has the highest energy barrier to date. Although well known, the performance of the 3D toric code has not been studied extensively. In this paper we propose efficient decoding algorithms for the 3D toric code on a cubic lattice with and without boundaries and report their performance for various quantum channels. We observe a threshold of $\ensuremath{\gtrsim}12%$ for the bit flip errors, $\ensuremath{\approx}3%$ for the phase flip errors, and 24.8% for the erasure channel. We also study the performance of the welded 3D toric code on the quantum erasure channel. We did not observe a threshold for the welded code over the erasure channel for the parameters we simulated.
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
This thesis is devoted to studying a class of quantum error-correcting codes — topological quantum codes. We explore the question of how one can achieve fault- tolerant quantum computation with topological codes. We treat quantum error-correcting codes not only as a compelling ingredient needed to build a quantum computer, but also as a useful theoretical tool in other areas of physics. In particular, we explore what insights topological codes can provide into challenging questions, such as the classification of quantum phases of matter. In this thesis, we focus on a family of topological codes — color codes, which are particularly intriguing due to the rich physics they display and their computational power. We start by introducing color codes and explaining their basic properties. Then, we show how to perform fault-tolerant universal quantum computation with three-dimensional color codes by transverse gates and code switching. We later compare the resource overhead of the code-switching approach with that of a state distillation scheme. We discuss how to perform error correction with the toric and color codes, as well as introduce local decoders for those two families of codes. By exploiting a connection between error correction and statistical mechanics we estimate the storage threshold error rates for bit-flip and phase-flip noise in the three-dimensional color code. We finish by showing that the color and toric code families in d dimensions are equivalent in a sense of local unitary transformations and explore implications of this equivalence.
We introduce an efficient decoder of the color code in d ≥ 2 dimensions, the Restriction Decoder, which uses any d -dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold p 2D ≈ 10.2 % on the square-octagon lattice is on a par with the toric code threshold on the square lattice.
Efficient color code decoders in d≥2d\geq 2d≥2 dimensions from toric code decoders
We introduce an efficient decoder of the color code in d≥2d\geq 2d≥2 dimensions, the Restriction Decoder, which uses any ddd-dimensional toric code decoder combined with a local lifting procedure to find a recovery operation. We prove that the Restriction Decoder successfully corrects errors in the color code if and only if the corresponding toric code decoding succeeds. We also numerically estimate the Restriction Decoder threshold for the color code in two and three dimensions against the bit-flip and phase-flip noise with perfect syndrome extraction. We report that the 2D color code threshold p2D≈10.2%p_{\textrm{2D}} \approx 10.2\%p2D​≈10.2% on the square-octagon lattice is on a par with the toric code threshold on the square lattice.